Equivalence in mapping of conductivity, elasticity and diffusion equations appears or disappears

摘要

Equivalence in mapping of diffusion and conductivity equations appears when diffusion and conductivity belong to the same universality class and disappears in the opposite case. For asymptotic long-time behavior one exponent of regular diffusion coefficient above the threshold D-infinity(e) proportional to (p - p(c))(mu-g) and two below the threshold D-infinity(e) proportional to (p - p(c))(-s-g) and D-infinity(e) proportional to (p - p(c))(s) are introduced, using the conductivity critical exponent (mu and s) and the Hall coefficient critical exponent R-e(p - p(c))(-g) if R-2/R-1 --> 1 and R-e(p - p(c))(2s) if R-2/R-1 --> infinity and p < p(c). In the first two cases equivalence between the diffusion and conductivity problems are disappear. A problem with this analogy lies in an intrinsic property that for percolation lattice the diffusion coefficient is not a self-averaging quantity unlike the conductivity. Analogy between conductivity and elasticity appears for scalar forces and disappears for vector forces. The latter leads to new equivalence between elasticity and conductivity in a magnetic field. The new formulae for a real part of dielectric constant in the static limit for p < p(c) and the low-frequency ac conductivity are determined due to the new critical exponent for regular diffusion. [References: 25]